What can the Cycles Geometry node 'Parametric' Socket be used for?
This output represents the "barycentric coordinates" in the rendered triangle. They can be used to  interpolate any data defined at the triangle's vertices.

In practice, however, as The Cycles Input Encyclopedia  explains: 


  It’s another one you probably won’t use very often, unless you’re trying to make an unusual wireframe effect, although we now have a specific node for that.


EDIT Here's an idea how you could use it: you can find the "center of mass" of the triangle - it is where all three coordinates have the same value. In practice you would need to work with an error threshold, and find a small triangle in the rendered triangle.

Another idea: Some software, such as Modo, have "Barycentric UV projection", you could probably emulate that, and use textures created in them.